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962 Part E Scattering Experiment

65.4.3 Excitation and Ionization

Excitation and ionization of inner shells of fast

projectile ions by the quasifree electrons of light tar-

gets (usually He or H

) have been identiﬁed and

studied. This process competes with excitation and

ionization by the target nucleus [65.118–120], and

special signatures must be sought to distinguish the

processes. In the case of excitation, the e–e ex-

citation populates states through the exchange part

of the interaction which is excluded for the nu-

clear excitation, and this has been used to separate

this mechanism [65.121]. For ionization, enhance-

ments of the ionization cross section above the

Born result for nuclear ionization [65.122], coinci-

dent charge exchange measurements [65.123], and

projectile [65.124] and recoil ion momentum spec-

troscopy [65.125, 126] have been used to distinguish

the two processes. Rapid development in the produc-

tion of good sources of beams of highly charged ions

(EBIS/T, ECR; see [65.127]) have made these stud-

ies possible. Continued study of this ﬁeld in heavy

ion storage rings is now achieving resolutions of

meV and opening broad new opportunities for data

of unprecedented high quality for electron-ion colli-

sions [65.128].

Recently, the ﬁrst kinematically complete exper-

iment on projectile ionization by quasifree elec-

trons was reported [65.129]. The observed features

are qualitatively similar to those found for ion-

ization of neutral target atoms by free electron

impact.

65.5 Some Exotic Processes

65.5.1 Molecular Orbital X-Rays

A typical time duration for an ion–atom encounter

is ∼ 10

−17

s, which is much shorter than Auger and

X-ray lifetimes, so that hard characteristic radiation

is emitted by the products long after the collision.

There remains, however, a small but ﬁnite proba-

biltiy that X-rays or Auger electrons can be emitted

during the collision, in which case the radiation

proceeds between the time-dependent molecular or-

bitals formed in the collision and reﬂects the time

evolution of the energies and transition strengths be-

tween the orbitals. Such molecular orbital X-radiation

(MOX) has now been observed in many collision sys-

tems [65.130] and is reviewed in [65.131–133]. MOX

spectra have been studied in total cross sections as

well as a function of impact parameter. In the lat-

ter case, oscillating structures in the MOX spectra are

seen [65.132, 134], due to the interference between

amplitudes for the emission of X-rays with the same

energy on the incoming and outgoing parts of the

trajectory.

The formation of transient molecular orbitals in

close collisions between highly charged ions pro-

vides opportunities for studying the electrodymamics

of very highly charged systems [65.135]. For exam-

ple, two uranium nuclei passing within one K-shell

radius of each other form a transient molecule

whose energy levels resemble those of an atom of

charge 184.

65.5.2 Positron Production

from Atomic Processes

Investigating the MOX interference patterns in such

exotic systems offers the interesting prospect of per-

forming spectroscopy on superheavy ions.

Reinhart et al. [65.136] predicted that, for such

highly charged species, the binding energy of the united

atom K-shell exceeds twice the rest mass energy of the

electron, and that if a K vacancy is either brought into

the collision or created during it, spontaneous electron-

positron pair production occurs (the decay of the charged

vacuum) with the electron ﬁlling the K-hole. However,

further analysis showed that the dominant mechanism

for positron production (other than those resulting from

the decay of nuclear excitations) likely results from the

dynamic time dependence of the ﬁelds during the col-

lision [65.137]. Experiments showed evidence for such

positron production in collisions at 6 MeV/u [65.138],

but reported sharp lines in the positron spectra were later

attributed to an error in the data analysis.

Present theories for the production of lepton pairs

in the close collision of two highly charged systems

predict that the cross section grows rapidly with col-

lision energy. Electrons produced in such a process

may end up in bound states on either collision partner,

and thus represent a new charge changing mechanism.

At highly relativistic velocities, the cross section for

this process exceeds that for any other charge chang-

ing process. In a heavy ion collider, such as RHIC,

Part E 65.5

Ion–Atom Collisions – High Energy References 963

this process could limit the ultimate storage time for

the counter-propagating beams, since charge-exchanged

ions are lost. The cross section has been measured re-

cently by Vane et al. [65.139], for 6.4 TeV S on several

targets (the highest energy ion–atom collision experi-

ment performed to date), and are in good agreement

with theory [65.140]. The bound state capture has been

measured at lower energy by Belkacem et al. [65.141],

with similarly good agreement. The extension of ion–

atom collisions to such extreme velocities has opened

the ﬁeld for the study of processes not even imagined

a short time ago.

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Part E 65

966 Part E Scattering Experiment

65.141 A. Belkacem, H. Gould, B. Feinberg, R. Bossing-

ham, W. E. Meyerhof: Phys. Rev. Lett. 71,1514

(1993)

65.142 W. König, F. Bosch, P. Kienle, C. Kozhukarov,

H. Tsertos, E. Berdermann, S. Muckler, W. Wag-

ner: Z. Phys. A. 328,129(1987)

Part E 65

967

Reactive Scatt

66. Reactive Scattering

This chapter presents a résumé of the methods

commonly employed in scattering experiments

involving neutral molecules at chemical energies,

i. e., less than about 10 eV. These experiments

include the study of intermolecular potentials,

the transfer of energy in molecular collisions, and

elementary chemical reaction dynamics. Closely

related material is presented in Chapts. 35, 37,

and 38 as well as in other chapters on quantum

optics.

66.1 Experimental Methods ......................... 967

66.1.1 Molecular Beam Sources............. 967

66.1.2 Reagent Preparation.................. 968

66.1.3 Detection of Neutral Products ..... 969

66.1.4 A Typical Signal Calculation......... 971

66.2 Experimental Conﬁgurations ................ 971

66.2.1 Crossed-Beam Rotatable Detector 971

66.2.2 Doppler Techniques ................... 973

66.2.3 Product Imaging ....................... 973

66.2.4 Laboratory to Center-of-Mass

Transformation ......................... 975

66.3 Elastic and Inelastic Scattering ............. 976

66.3.1 The Differential Cross Section ...... 976

66.3.2 Rotationally Inelastic

Scattering................................. 977

66.3.3 Vibrationally Inelastic

Scattering................................. 977

66.3.4 Electronically Inelastic

Scattering................................. 978

66.4 Reactive Scattering .............................. 978

66.4.1 Harpoon and Stripping

Reactions ................................. 978

66.4.2 Rebound Reactions ................... 979

66.4.3 Long-lived Complexes ............... 979

66.5 Recent Developments........................... 980

References .................................................. 980

66.1 Experimental Methods

66.1.1 Molecular Beam Sources

The development of molecular beam methods in the

past two decades has transformed the study of chem-

ical physics [66.1]. Supersonic molecular beam sources

allow one to prepare reagents possessing a very nar-

row velocity distribution with very low internal energies,

ideal for use in detailed studies of intermolecular interac-

tions. Early experiments generally employed continuous

beam sources, but in recent years intense pulsed beam

sources have come into common usage [66.2]. The ad-

vantages of pulsed beams primarily arise from the lower

gas loads associated with their use, hence reduced de-

mands on the pumping system. If any component of

the experiment is pulsed (pulsed laser detection, for ex-

ample) then considerable advantage may be obtained

by also pulsing the beam. Although the theoretical

descriptions of pulsed and continuous expansions are

essentially equivalent, in practice some care is required

in employing pulsed beams because the temperature and

velocity distributions may change dramatically through

the course of the pulse. Free jet expansions are super-

sonic because the dramatic drop in the local temperature

in the beam is associated with a drop in the local speed

of sound. A detailed description of the supersonic ex-

pansion may be found in [66.3–5]. In practice, many of

the detailed features associated with a supersonic expan-

sion may be ignored and one may assume an isentropic

expansion into the vacuum. For an isentropic nozzle ex-

pansion of an ideal gas, the maximum terminal velocity

is given by

max



, (66.1)

where, for an ideal gas, the heat capacity is



γ −1



(66.2)

R is the gas constant, m is the molar molecular mass,

the temperature in the stagnation region, and γ the

Part E 66

968 Part E Scattering Experiment

heat capacity ratio. For ideal gas mixtures, and assuming

independent of temperature for the range encountered

in the expansion, one may use





−1



R ,

(66.3)

and the average molar mass

m =



, (66.4)

where X

is the mole fraction of component i, to obtain

an estimate of the maximum velocity for a mixture:

max



m . (66.5)

By seeding heavy species in light gases one may acceler-

ate them to superthermal energies. Supersonic beams are

characterized by the speed ratio, i. e., the mean velocity

divided by the velocity spread:

S ≡

√

2kT/m

(66.6)

where T is the local translational temperature, or by the

Mach number

M ≡

√

γkT/m

(66.7)

For the purpose of order-of-magnitude calculations, the

number density on axis far from the nozzle may be

estimated as

n ≈ n

(

d/x

)

, (66.8)

where n

is the number density in the stagnation region,

d is the nozzle diameter, and x is the distance from the

nozzle. The number density versus speed distribution of

a nozzle beam is well described as a Gaussian charac-

terized by the speed ratio S and a parameter α =v

/S,

where v

is the most probable velocity:

n(v) = v

exp



−

(

v/α −S

)



(66.9)

Cooling efﬁciencies for the various internal degrees of

freedom correlate with the efﬁciencyof coupling of these

modes with translation, hence they vary widely. Coup-

ling of modes A and B is expressed by the collision

number Z

A−B

≡ Zτ

A−B

, (66.10)

where τ

A−B

is the bulk relaxation time, and Z the colli-

sion frequency. This represents the number of collisions

between effective inelastic events. Typical values are

Table 66.1 Collision numbers for coupling between dif-

ferent modes. V, R, T refer to vibrational, rotational, and

translational energy, respectively. Each entry is the typical

range of Z

A−B

V R T

V 10

0.5−3

3−4

5−6

R 10

0−1

2−3

summarized in Table 66.1. R–T coupling is relatively

efﬁcient, while V–T coupling is quite inefﬁcient, so that

vibrational excitation may not be effectively cooled in

the expansion.

66.1.2 Reagent Preparation

Molecular beam methods may be used in conjunc-

tion with a variety of other techniques to prepare

atoms or molecules in excited or polarized initial states

(Chapt. 46), to generate unstable molecules or radi-

cals [66.6,7] or to produce beams of refractory materials

such as transition metals or carbon [66.8,9]. Some of the

common techniques are outlined below. Optical pump-

ing of atoms to excited electronic states is a useful means

of reagent preparation, and this topic is presented in

detail in Chapt. 10. This technique further allows one,

using polarized lasers, to explore the inﬂuence of angular

momentum polarization in the reagents on the colli-

sion dynamics [66.10]. Most of these studies have been

performed using alkali and alkaline earth metals since

there exist strong electronic transitions and convenient

narrow-band visible lasers suitable for use with these

systems. Laser excitation may also be used to generate

vibrationally excited molecules in their ground elec-

tronic states. The techniques employed include direct

IR excitation using an HF chemical laser [66.11], pop-

ulation depletion methods [66.12] and various Raman

techniques [66.13].

Metastable atoms may also be prepared by laser

photolysis of a suitable precursor. O





preparation is

readily prepared by photolysis of ozone or N

O, for ex-

ample [66.14]. Alternatively, rf or microwave discharges

may be used to produce metastable species or reactive

atoms or radicals [66.15]. These techniques may also

be used to prepare ground state atoms; for example, hot

H atom beams are frequently produced by photolysis

of HI or H

S [66.16]. Such atomic or molecular radical

beams may also be generated by pyrolysis in the nozzle.

In this case care must be taken to minimize recombina-

tion through careful choice of the temperature, nozzle

geometry, and transit time through the heated region.

Part E 66.1

Reactive Scattering 66.1 Experimental Methods 969

Beams of refractory materials are now commonly gen-

erated using laser ablation sources [66.8, 9]. Typically

these employ a rod or disk of the substrate of interest

which is simultaneously rotated and translated to pro-

vide a fresh surface for ablation at each laser pulse.

A laser beam is focused on the substrate and timed to

ﬁre just as a carrier gas pulse passes over. Laser power

and wavelength must be optimized for a given substrate.

Lasers operating in the IR, visible, and UV have all

been employed. Aligned or oriented molecules have

been prepared using multipole focussing [66.17, 18],

and more recently using strong electric ﬁelds (“brute

force”) [66.19]. In the former case, speciﬁc quantum

states are focused by the ﬁeld. In the latter case, so called

pendular states are prepared from the low rotational lev-

els of molecules possessing large dipole moments and

small rotational constants. The ability to orient these

molecules can be estimated on the basis of the Stark

parameter ω = µE/B,whereµ is the dipole moment,

E the electric ﬁeld strength, and B the rotational con-

stant. Orientation is feasible for low rotational levels of

molecules when the Stark parameter is on the order of

10 or higher [66.19].

66.1.3 Detection of Neutral Products

Broadly speaking, detection of neutral molecules is

accomplished either by optical (spectroscopic) or

nonoptical techniques. Nonoptical methods usually in-

volve nonspeciﬁc ionization of neutral particles, most

commonly by electron impact, followed by mass se-

lection and ion counting. Thermal detectors such as

cryogenic bolometers are also ﬁnding widespread ap-

plication in molecular beam experiments owing to their

remarkable sensitivity [66.20]. In general, optical meth-

ods may rely on resonant or nonresonant processes,

hence they may or may not enjoy quantum state selectiv-

ity. Both photoionization and laser-induced ﬂuorescence

methods are now in common usage, usually in appli-

cations where quantum state resolved information is

desired. The advantage of nonoptical methods is pri-

marily one of generality: all neutral molecules may be

detected, and branching into different channels readily

measured. Quantum state resolution is more difﬁcult to

achieve using nonoptical detection methods, but both

vibrationally- and rotationally-resolved measurements

have been obtained by these means [66.21,22].

The primary advantage of spectroscopic detection

is the aforementioned possibility of quantum state

speciﬁcity. Another unique opportunity afforded by

spectroscopic probes is the measurement of product

aligment and orientation. In addition, in some cases

background interference may be reduced or eliminated

using state-speciﬁc probes, thereby affording enhanced

signal-to-noise ratios.

Nonoptical Techniques

Detectors based on nonspeciﬁc ionization remain the

most commonly used in molecular beam experiments,

owing to the ease of subsequent mass selection, and

the convenience and sensitivity of ion detection. Sur-

face ionization is a sensitive means of detecting alkali

atoms and other species exhibiting low ionization poten-

tials [66.23]. Surface ionization occurs when a neutral

atom or molecule with a low ionization potential sticks

on a surface with a high work function and is subse-

quently desorbed. Typically these detectors employ a hot

platinum or oxidized tungsten wire or ribbon for forma-

tion and subsequent desorption of the ions, which is

surrounded by an ion collector. They are very efﬁcient

for the detection of alkali atoms and molecules whose

ionization potentials are

 6eV.

All neutral gas molecules may be ionized by colli-

sion with energetic electrons, and electron beam ionizers

may be produced that couple conveniently to quadrupole

mass spectrometers [66.24]. Collision ofa molecule with

a 100–200 eV electron leads predominantly to formation

of the positive ion and a secondary electron. Other pro-

cesses also occur and can be very signiﬁcant: doubly

or triply charged ions may be formed and, importantly,

molecules can fragment yielding many daughter ions

in addition to the parent ion. These fragmentation pat-

terns vary with different molecules, and may further

show a strong dependence on molecular internal energy,

so particular care must be taken to determine the role

of these phenomena in each particular application. It is

often necessary to record data for the parent ion and

daughter ions for a given product channel and compare

them to eliminate contributions arising from cracking

of the parent molecule or other species [66.25]. Elec-

tron impact ionization probabilities for most species

exhibit a similar dependence on electron energy, ris-

ing rapidly from the ionization potential to a peak at

80–100 eV, then fallingmore slowly with increasing col-

lision energy. The ionization cross section for different

species scales with molecular polarizability according

to a well-established empirical relation [66.26]:

ion

= 36

√

α −18 , (66.11)

where σ

ion

is in Å

and α, the molecular polarizability,

is in Å

. This relation can be used to estimate branching

ratios in the absence of any other means of calibrating

Part E 66.1

970 Part E Scattering Experiment

the relative contributions of two different channels. The

ionization rate is given by

d[M

]

= I

σ[M] , (66.12)

where I

is the electron beam intensity, typically

10 mA/cm

or 6 × 10

electrons/cm

s, and [M] is the

number density of molecules M in the ionizer. If one

assumes an ionization cross section σ

ion

of 10

−16

for collision with 150 eV electrons (a typical value

for a small molecule), the ionization probability for

molecules residing in the ionizer is then

d[M

]

[M]

= I

σ =6×10

×10

−16

= 6s

−1

(66.13)

However, product molecules arriving in the detector are

not stationary. Typically, product velocities are on the

order of 500 m/s. If the ionization region has a length

of 1 cm, the residence time τ of a product molecule is

on the order of 2×10

−5

s. Consequently, the ionization

probability of product molecules passing through the

ionizer is

d[M

]

[M]

= 2×10

−5

×6= 1.2×10

−4

. (66.14)

Although this does not appear very efﬁcient (indeed, it

is 4 orders of magnitude less so than surface ionization),

nevetheless, if the background count rate is sufﬁciently

low, then good statistics may be obtained with signal lev-

els as low as 1 Hz. Thus, for detection based on electron

impact ionization, a key factor determining the sensitiv-

ity of the experiment is the background count rate at the

masses of interest.

Spectroscopic Detection

Spectroscopic detection methods usually involve either

laser-induced ﬂuorescence (LIF) or resonant photoion-

ization (REMPI)(Chapt.44). Alternative techniques

such as laser-induced grating methods and nonresonant

VUV photoionization are also being applied to scatter-

ing experiments. Essential to the use of spectroscopic

methods for reactive scattering studies is an understand-

ing of the spectrum of the species of interest. This may

be challenging for many reactive systems because the

products may be produced in highly excited vibrational

or electronic states that may not be well characterized.

Additional spectroscopic data may be required. Franck–

Condon factors are necessary to compare the intensities

of different product vibrational states, while a calibration

of the relative intensities of different electronic bands re-

quires a measure of the electronic transition moments.

In some cases, one must include the speciﬁc dependence

of the electronic transition moment on the internuclear

distance by integrating over the vibrational wave func-

tion. Populations corresponding to different rotational

lines may be compared after the appropriate correc-

tion, which is represented by the Hönl–London factors,

only for isotropic irradiation and detection. This is cer-

tainly not the case for most laser-based experiments.

Generally, the detailed dependence of the excitation and

detection on the relevant magnetic sublevels must be

considered [66.27–29].Cautionisrequiredinusingany

spectroscopic method involving a level that is predis-

sociated. This may lead to a dramatic decrease in the

associated ﬂuorescence or photoionization yield if the

predissociation rate approaches or exceeds the rate of

ﬂuorescence or subsequent photoionization. An impor-

tant question in any experiment based on spectroscopic

detection is whether product ﬂux or number density is

probed. This question is considered in detail in several

articles [66.13,30]. It depends on the lifetime of the state

that is probed, the relative time that the molecule is ex-

posed to the probe laser ﬁeld, and its residence time in

the interaction region. Saturation phenomena are also

important, yet not necessarily easily anticipated. Com-

plete saturation does not readily occur because excitation

in the wings of the laser beam proﬁle becomes more sig-

niﬁcant as the region in the center of the beam becomes

saturated [66.31].

LIF is currently the most widely used spectroscopic

technique in inelastic and reactive scattering experi-

ments [66.27, 32, 33]. It has been used to measure

state-resolved total cross sections [66.34] and differ-

ential cross sections in electronic [66.35], vibrational

and rotationally inelastic scattering [66.12]aswellas

reactive scattering [66.36].

With the development of high-power tunablebreak

lasers and the discovery of useful photoionization

schemes, REMPI is becoming a more general tech-

nique [66.37,38]. REMPI has the advantages associated

with ion detection, namely considerable convenience in

mass selection and efﬁcient detection, in addition to

the capability for quantum state selectivity. Disadvan-

tages associated with REMPI arise primarily from higher

laser power employed compared with LIF. Caution is re-

quired in attempting to extract quantitative information

from REMPI spectra if one or several of the steps in-

volved in the ionization process are saturated. This is

of particular concern at the high laser powers neces-

sary for multiple photon transitions. An alternative to

direct photoionization involves excitation of products to

metastable Rydberg states, followed by ﬁeld ionization

Part E 66.1

Reactive Scattering 66.2 Experimental Conﬁgurations 971

some distance from the interaction region. This tech-

nique has the advantage of very low background and is

capable of extraordinary time-of-ﬂight resolution. Re-

markable results have recently been obtained for the

reaction D +H

using this method [66.39]. Photoion-

ization techniques are becoming more widely used in

scattering experiments as the basis for product imaging

detection schemes discussed below.

66.1.4 A Typical Signal Calculation

For a crossed-beam system in which a beam of atoms A

collides with a beam of molecules B yielding products

C and D, the rate of formation of C is given by

= n

g∆V , (66.15)

where n

and n

are the number densities of the respec-

tive reagents at the interaction region, σ

is the reaction

cross section, g the magnitude of the relative velocity be-

tween the reactants, and ∆V the volume of intersection

of the beams. For a typical experiment employing con-

tinuous supersonic beams, the number densities of the

atomic and molecular reactants are ∼ 10

–10

−3

and the scattering volume 10

−2

.Forg =10

cm/s

and σ

= 10

−15

, the rate of product formation

/dt = 10

molecules/s. The kinematics and en-

ergetics of the reaction then determines the range of

laboratory angles into which the products scatter, and

the magnitude of the scattered signal.

If the products scatter into 1 sr of solid angle, and

the detector aperture is 3 × 10

−3

sr (roughly 1 degree

in both directions perpendicular to the detector axis),

then the detector receives 3 × 10

product molecules/s.

Given the detection probability obtained above,

3600 product ions/s are detected. This is adequate to

obtain very good statistics in a short time as long as the

background count rate is not considerably higher.

For a nonspeciﬁc detection technique, such as

electron bombardment ionization coupled with mass

ﬁltering, it is necessary to use ultrahigh vacuum



−10

torr



in the detector region to minimize interfer-

ence from background gases. The residual gases are then

primarily H

and CO, with number densities on the order

of 10

. Differential pumping stages, each of which

may reduce the background by 2 orders of magnitude,

are generally used to lower the background from gases

whose partial pressures are lower than the ultrahigh

vacuum limit of the detector chamber. However, this

differential pumping helps only for those molecules that

do not follow a straight trajectory through the detector.

The contribution from the latter is given by



4πx

, (66.16)

where n is the number density of molecules effusing

from an oriﬁce of area A,andn



is their number density at

a distance x on axis downstream. For a distance of 30 cm

and a main chamber pressure of 3 × 10

−7

torr, this corres-

ponds to a steady state density of 10

molecules/cm

the ionizer, a reduction of 6 orders of magnitude. Three

stages of differential pumping are thus the maximum

useful under these conditions, since the primary source

of background is then molecules following a straight tra-

jectory from the main chamber. A liquid helium cooled

surface opposite the detector entrance may then be use-

ful to minimize scattering of background molecules into

the ionizer.

66.2 Experimental Conﬁgurations

66.2.1 Crossed-Beam Rotatable Detector

The conﬁguration illustrated in Fig. 66.1 represents

a standard now widely used [66.24], usually with two

continuous beams ﬁxed at 90 degrees. The molecular

beam sources are differentially pumped and collimated

to yield an angular divergence of about 2 degrees.

The beams cross as close as possible to the nozzles,

with a typical interaction volume of 3 mm

. Scattered

products pass through an aperture on the front of the

detector, thence through several stages of differen-

tial pumping before reaching the ionizer. Ions formed

by electron impact on the neutral products are then

extracted into a quadrupole mass spectrometer with as-

sociated ion counter. A chopper wheel is generally used

at the entrance to the detector to provide a time ori-

gin for recording time-of-ﬂight spectra. Pseudorandom

sequence chopper disks provide optimal counting statis-

tics while maintaining a high duty cycle (50%) [66.40].

The detector may be rotated about the interaction region,

typically through a range of 120

◦

or so, allowing one to

examine products scattered at a range of laboratory an-

gles. In addition to time-of-ﬂight detection, one of the

beams may be gated on and off for background sub-

Part E 66.2